Day 12 in Mat430

Last time: Limits

Next time: Exam 1

Today:

Announcements
- Homework p. 42 due today: #1, 3, 4, 6, 8a
- Exam 1 will be this Wednesday, 2/10, and will cover through continuity (section 17).
  - Structure of the exam
    - Your one homework set will be part of the exam
    - It will be primarily computational and conceptual
    - I'll give you a problem or two that may require more time (e.g. a mapping problem) to hand in later.
- Today's activity, and new assignment:
  Mon 2/8 15-17
  
  Prepare for exam. The exam will cover through section 15.
  No new reading or problems.
Section 15: Theorems on Limits
- (Limits exist) iff (limits of components exist)
- Given that limits of f and g exist at z₀. Then limits of sums, products, and quotients exist, as expected (usual caveat for quotients -- denominator doesn't go to zero)
Section 16: The point at infinity
- We extend the complex plane by adding the point at infinity!?
- Riemann sphere (p. 49), and the stereographic projection between the complex plane and the Riemann sphere.
- An ${\epsilon}$ neighborhood of infinity is the set of points ${|z|>1/\epsilon}$
- Theorem (p. 49):
  
  ${\lim_{z\rightarrow{z_0}}f(z)=\infty}$ iff ${\lim_{z\rightarrow{z_0}}\frac{1}{f(z)}=0}$
  
  ${\lim_{z\rightarrow{\infty}}f(z)=w_0}$ iff ${\lim_{z\rightarrow{\infty}}f(\frac{1}{z})=w_0}$
  
  ${\lim_{z\rightarrow{\infty}}f(z)=\infty}$ iff ${\lim_{z\rightarrow{\infty}}\frac{1}{f(1/z)}=0}$
- I'd like to see if we can make the correspondence between points and the plane and points on the sphere explicit, by coming up with an equation for the one-to-one correspondence.
Section 17: Continuity
- Continuity of Polynomials
- Continuity of Rational functions
- Continuity of compositions of continuous functions
- and others....

Website maintained by Andy Long. Comments appreciated.

${\lim_{z\rightarrow{z_0}}f(z)=\infty}$	iff	${\lim_{z\rightarrow{z_0}}\frac{1}{f(z)}=0}$
${\lim_{z\rightarrow{\infty}}f(z)=w_0}$	iff	${\lim_{z\rightarrow{\infty}}f(\frac{1}{z})=w_0}$
${\lim_{z\rightarrow{\infty}}f(z)=\infty}$	iff	${\lim_{z\rightarrow{\infty}}\frac{1}{f(1/z)}=0}$